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QC Strategies - Estimation
Now that we’ve talked about the format a bit. We’re going to have a couple lessons on just basic strategies for the quantitative comparison questions. And the number one strategy is, don’t do more calculations than necessary, your job is to compare, not to calculate, you only calculate if you absolutely have to. All you have to do is comparison.
Often, we don’t need to figure out exact values of quantities in A and B, individually. We just need to determine which one is bigger. Approximation can be an excellent method for making a quick comparison. So, here’s an elementary math problem.
This is actually a little bit easier than what you’d actually see on the GRE. But pause the video and then we’ll talk about this. So I realize we haven’t covered geometry yet, so you may be a little rusty with the formulas. I’ll remind you that the circumference of a circle is this magic number pi times the diameter.
The, another formula for the circumference, incidentally, is 2pi r, where the, the radius, of course, is half the diameter. So again, we’ll cover this when we have our video on circles in the geometry section much later on. So if you didn’t remember that right now, don’t worry about that. The important point is, suppose we had to this comparison.
So we have, we know the diameter’s 4, we have to compare 4pi to 12. How would we do this? Well, the mistake would be, to try and remember, now wait a second, pi is 3.14, let me reach for the calculator and do 4 times 3.14. Reaching for the calculator in a question like this would be a huge trap. You don’t have to use a calculator at all.
And in fact, it, you almost never have to touch the calculator on the quantitative comparison questions. Think about it this way. I’m just gonna say, what is pi? How big is pi? Well, I’m gonna write it this way.
Pi is 3, with a little plus sign, and what I mean by that is, pi is a number slightly bigger than 3. I don’t even care how much, it’s slightly bigger than 3. Now, now that I know this, what can I do with this? Well, think about this, 4 times something slightly bigger than 3 is gonna be bigger than 4 times 3.
So, quantity A has to be bigger because something slightly bigger than 3 is larger than exactly 3. So, that’s why A has to be bigger. Again, the exact value, the exact calculation, absolutely irrelevant to answering this question. Often it’s enough to figure out whether something is bigger or smaller than a particular integer or a particular round-number multiple.
We can also compare two quantities part-wise. Suppose each quantity has two parts, two things added together. If each part of one column is bigger than the corresponding part of the other column, then the quantity with the bigger parts must be bigger. With that in mind, think about this question. Pause the video and then we’ll talk about this.
Okay, I realize I’m being very mean here, I’m throwing fractions at you. It may be that you haven’t dealt with fractions in awhile. Don’t worry, we have a whole set of videos coming up on what you can do with fractions, what you can’t do with fractions. If you’re rusty with fractions right now, don’t worry about that.
But the big idea is, it would be a huge mistake to try and figure out exactly what does Quantity A equal, exactly what does Quantity B equal. It would be insane to find the common denominators, to find the exact answers, that’s not what the question’s about. And the people who actually do those calculations, let me figure out exactly what Quantity A equals, let me figure out exactly what Quantity B equals.
The people that do those calculations have fallen into a trap. Because that’s not what the question is about. Think about it this way. Look at that first fraction in Quantity A and the first fraction in Quantity B. Let’s look at that fraction 13 over 27. Well certainly, if we make the denominator bigger, that makes the fraction smaller.
So 13 over 27 has to be greater than 13 over 28. Well if we make the numerator smaller, that makes the fraction even smaller. So 12 over 28. Well 12 over 28 simplifies to three-sevenths. So this tells us that 13 over 27 is larger than three-sevenths so that piece is bigger.
Now, look at the other piece, two-fifths and 41 over 97. Well, 41 and 97, those are awfully close to 40 and 100. Let’s think about this carefully. Let’s start with 41, 97 and again, increase the denominator to 100. So that makes the fraction a little bit smaller when we increase the denominator. Then decrease the numerator, that makes the fraction even smaller.
And of course, 40 over 100 equals two-fifths. So each piece of Quantity B is bigger than the corresponding piece of Quantity A. And so that means what ever the sum of Quantity B is, it has to be bigger than the sum of Quantity A. So notice that we just did a piece-wise comparison with approximations. The actual, the exact, values of the quantities are irrelevant to our calculation here.
In other words, you do not get sucked into a long calculation, you’re merely doing a comparison. Here’s another practice problem. Pause the video and then we’ll talk about this. Okay, this one is a screaming trap.
So many people, if they ran into this question on the GRE, they would reach for the calculator. And they’d be typing all those numbers in. Big, big mistake. Look at these numbers. For example, 52.8% of 5,929 in Quantity A.
Well, notice that percent, 32.8. That’s slightly less than 33.3%. And, of course, 5,929 is slightly less than 6,000. So, I’m just gonna write it like this. Slightly less than 33.3% of slightly less than 6,000. And 33.3% is just a third, so a third of 6,000 would be 2,000, so it means that overall, whatever this answer is gonna be, it’s gonna be something slightly less than 2,000.
Now look at the other one, 41.6% of something a little over 5,000. So I’m going to write this again as something a little over 40% of something a little over 5,000. Well 40% of 5,000 would be 2,000. And this answer would equal something slightly larger than 2,000.
So, again, I don’t care about the exact values. All I care about is the comparison. And notice that this was set up to have a nice, easy comparison. So that we could just see, one’s smaller than 2000, one’s larger than 2000. That means that the answer has to be B. Now you may think, how would I know to look at it this way?
This is really the trick of the QCs. And I’m trying to show you as many patterns as possible, so you start to think in terms of these patterns. What are ways that I can look at the problem and do some kind of simplification so that the comparison becomes easy? That’s the real question on the QCs.
If you get sucked into the long calculation, you have fallen into the trap, and you’re doing something much harder than you have to. Notice that if both columns contain numbers, then there must be some definitive relationship between the two quantities, in other words D can’t possibly be the answer.
And, in fact, that was true in, in all three examples that I’ve had here. Even just glancing at the problem, it should be immediately obvious D can’t be the answer. Because if you have two quantities that are purely numerical, then must be true that either one is bigger than the other, or that the two are equal. Also notice that on the two previous problems, the problems looked as if they would involve long tedious calculation.
But that’s absolutely not what the problems were really about. And this is such an important idea. Instead, both problems were really testing the students’ ability to look at the problem in the right way, to dissect it in a way that would make the relationship easy to see. This is quite typical of what the GRE will ask students.
And it’s so important to realize this. The GRE Quant section is not about long, detailed calculations. The, the GRE math section is about, can you look at the problem and break it apart in the right way or simplify it in the right way or dissect it in the right way so that it becomes easy to answer? That’s the mindset you should be in when you’re doing GRE math.
In summary, the GRE does not care about the ability to do long, detailed calculations. On the QC questions, our job is to make quick comparisons. Estimation is a, is much more common on the quantitative comparison questions than are precise calculations. You might estimate on more than half of the quantitative comparison questions, it’s actually very, very rare to have a quantitative comparison question on which you have to do a precise calculation.
So some of the tricks I mentioned doing part-wise comparisons, that’s a strategy that comes up sometime, and then rounding things to, round numbers, doing comparisons to nice round numbers. These are just patterns to keep in mind. And finally, the most important skill on many QC questions is knowing how to look at the problem, and how to dissect it, so that the comparison is simple.
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